|Copyright||(c) 2011 diagrams-core team (see LICENSE)|
|License||BSD-style (see LICENSE)|
Diagrams defines the core library of primitives forming the basis of an embedded domain-specific language for describing and rendering diagrams.
Transform module defines generic transformations
parameterized by any vector space.
- data u :-: v = (u :-* v) :-: (v :-* u)
- (<->) :: (HasLinearMap u, HasLinearMap v) => (u -> v) -> (v -> u) -> u :-: v
- linv :: (u :-: v) -> v :-: u
- lapp :: (VectorSpace v, Scalar u ~ Scalar v, HasLinearMap u) => (u :-: v) -> u -> v
- data Transformation v = Transformation (v :-: v) (v :-: v) v
- inv :: HasLinearMap v => Transformation v -> Transformation v
- transp :: Transformation v -> v :-: v
- transl :: Transformation v -> v
- dropTransl :: AdditiveGroup v => Transformation v -> Transformation v
- apply :: HasLinearMap v => Transformation v -> v -> v
- papply :: HasLinearMap v => Transformation v -> Point v -> Point v
- fromLinear :: AdditiveGroup v => (v :-: v) -> (v :-: v) -> Transformation v
- basis :: forall v. HasLinearMap v => [v]
- dimension :: forall a. HasLinearMap (V a) => a -> Int
- onBasis :: forall v. HasLinearMap v => Transformation v -> ([v], v)
- listRep :: HasLinearMap v => v -> [Scalar v]
- matrixRep :: HasLinearMap v => Transformation v -> [[Scalar v]]
- matrixHomRep :: HasLinearMap v => Transformation v -> [[Scalar v]]
- determinant :: (HasLinearMap v, Num (Scalar v)) => Transformation v -> Scalar v
- avgScale :: (HasLinearMap v, Floating (Scalar v)) => Transformation v -> Scalar v
- class (HasBasis v, HasTrie (Basis v), VectorSpace v) => HasLinearMap v
- class HasLinearMap (V t) => Transformable t where
- newtype TransInv t = TransInv t
- translation :: HasLinearMap v => v -> Transformation v
- translate :: (Transformable t, HasLinearMap (V t)) => V t -> t -> t
- scaling :: (HasLinearMap v, Fractional (Scalar v)) => Scalar v -> Transformation v
- scale :: (Transformable t, Fractional (Scalar (V t)), Eq (Scalar (V t))) => Scalar (V t) -> t -> t
Invertible linear transformations
(v1 :-: v2) is a linear map paired with its inverse.
Create an invertible linear map from two functions which are assumed to be linear inverses.
Apply a linear map to a vector.
General (affine) transformations, represented by an invertible linear map, its transpose, and a vector representing a translation component.
By the transpose of a linear map we mean simply the linear map corresponding to the transpose of the map's matrix representation. For example, any scale is its own transpose, since scales are represented by matrices with zeros everywhere except the diagonal. The transpose of a rotation is the same as its inverse.
The reason we need to keep track of transposes is because it turns out that when transforming a shape according to some linear map L, the shape's normal vectors transform according to L's inverse transpose. This is exactly what we need when transforming bounding functions, which are defined in terms of perpendicular (i.e. normal) hyperplanes.
For more general, non-invertable transformations, see
|HasLinearMap v => Monoid (Transformation v)|
|HasLinearMap v => Semigroup (Transformation v)|
Transformations are closed under composition;
|HasLinearMap v => HasOrigin (Transformation v)|
|HasLinearMap v => Transformable (Transformation v)|
|(HasLinearMap v, (~) * v (V a), Transformable a) => Action (Transformation v) a|
Transformations can act on transformable things.
|type V (Transformation v) = v|
Get the transpose of a transformation (ignoring the translation component).
Drop the translational component of a transformation, leaving only the linear part.
Apply a transformation to a vector. Note that any translational component of the transformation will not affect the vector, since vectors are invariant under translation.
Apply a transformation to a point.
Create a general affine transformation from an invertible linear transformation and its transpose. The translational component is assumed to be zero.
Get the matrix equivalent of the basis of the vector space v as a list of columns.
Get the dimension of an object whose vector space is an instance of
HasLinearMap, e.g. transformations, paths, diagrams, etc.
Get the matrix equivalent of the linear transform, (as a list of columns) and the translation vector. This is mostly useful for implementing backends.
Convert a `Transformation v` to a matrix representation as a list of column vectors which are also lists.
Convert a `Transformation v` to a homogeneous matrix representation. The final list is the translation. The representation leaves off the last row of the matrix as it is always [0,0, ... 1] and this representation is the defacto standard for backends.
The determinant of a
Compute the "average" amount of scaling performed by a transformation. Satisfies the properties
avgScale (scaling k) == k avgScale (t1 <> t2) == avgScale t1 * avgScale t2
The Transformable class
HasLinearMap is a poor man's class constraint synonym, just to
help shorten some of the ridiculously long constraint sets.
Type class for things
t which can be transformed.
TransInv is a wrapper which makes a transformable type
translationally invariant; the translational component of
transformations will no longer affect things wrapped in
|Eq t => Eq (TransInv t)|
|Ord t => Ord (TransInv t)|
|Show t => Show (TransInv t)|
|Monoid t => Monoid (TransInv t)|
|Semigroup t => Semigroup (TransInv t)|
|Wrapped (TransInv t)|
|VectorSpace (V t) => HasOrigin (TransInv t)|
|Transformable t => Transformable (TransInv t)|
|Qualifiable a => Qualifiable (TransInv a)|
|Traced t => Traced (TransInv t)|
|Enveloped t => Enveloped (TransInv t)|
|Rewrapped (TransInv t) (TransInv t')|
|type Unwrapped (TransInv t) = t|
|type V (TransInv t) = V t|
Vector space independent transformations
Most transformations are specific to a particular vector space, but a few can be defined generically over any vector space.
Create a uniform scaling transformation.